4 Answers
4

"See the thing is, Harry, there's some wizards, like the Malfoy family,
who think they're better than everyone else because they're what
people call 'pure blood' .... And it's codswallop to boot. 'Dirty
blood'! Why there isn't a wizard alive today that's not half-blood or
less."

So the terms pure blood and half blood are mostly terms used by people who believe in the validity of these false monikers. And since Harry and the rest of the Weasley's never did subscribe to these ideas then it'd be safe to say "neither".

@MeatTrademark - It's obvious from Hagrid's comment that they are simply labels used only by certain people. And that they're false labels as well, since there are no pure blooded wizards anymore.
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System DownJan 3 '14 at 19:38

I understand your point, but just ignoring the question is no fun. The question does say "what do you think?" So I threw in my 2 cents for whatever it's worth. Besides, there is no "real" answer unless pure-blood is defined with strict parameters on generational drift from half-blood.
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Meat TrademarkJan 3 '14 at 19:48

Since they are both magical, the children should be pure-blood. Doesn't one parent need to be a muggle for a half-blood and both to be a "Mudblood"? (Sorry for dropping the M-bomb there...)

Lily was mugggle-born. He is a half-blood (or what would be more accurately described as 3/4-blood but the pure-blood haters gotta hate), therefore Harry and Ginny would technically have half-bloods because of his mother. It depends on how many generation you use to define purity.

It should be noted that the first site referenced above states:

Harry Potter — pure-blood father, Muggle-born mother

then goes on to list the children as

Two magical parents, known Muggle ancestry

Albus Severus Potter — half-blood father, pure-blood mother

James Sirius Potter — half-blood father, pure-blood mother

Lily Luna Potter — half-blood father, pure-blood mother

Sorry the answer is so messy, but the case could be made for either answer. If your only criteria is neither parent being a Muggle, then yes they're pure. If you need to wait generations or distort your family tree / hide your ancestry (as the "accepted" pure-bloods like the Blacks or the Malfoys almost certainly did), then no, they're not pure-blood.

That's the longest wishy-washy answer I've ever given, and I apologize, but I assure you I started off strongly in favor of them being pure-blood and got muddled down by the "facts," if you could call them that. Best intentions... meh. I vote pure.
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Meat TrademarkJan 3 '14 at 19:51

From the 'pureblood' view, they would be half-blood, since they have muggle ancestry. Pureblood is defined as never having mixed with muggles (if you ask most purebloods), or not having any knowledge of muggle ancestry (if you ask everyone else).

In Germany itself, the Ahnenpass and Nuremberg Laws classified people as Jews if they descended from three or four Jewish grandparents. A person with one or two Jewish grandparents was a Mischling, a crossbreed, of "mixed blood".

As far as Harry's children, they were "polluted" blood by those same ideas, but not "mixed blod". JKR herself said:

A single Jewish grandparent 'polluted' the blood, according to their propaganda.